Robust Multigrid Method for the Efficient Solution of PDEs on Complicated Domains
Nadin Stahn (Univ. Zürich)
The finite element discretisation of elliptic boundary value problems on
complicated domains often results in a huge number of unknowns. Multigrid
methods have the potential to solve the arising systems of linear equations in
linear complexity with respect to the number of unknowns.
We employ overlapping composite finite elements for the
construction of a sequence of coarse-level discretisations setting up
a multigrid algorithm.
The convergence of this algorithm is
proved in the framework of geometric multigrid methods.
The idea is to adapt the general convergence theory to the specific
situation and to prove the so-called smoothing and approximation
property. The emphasis is on the robustness with respect to the
possibly degenerate geometry of the intersection of overlapping
triangles with the domain. As a
consequence the matrix entries which correspond to nodes lying
(essentially) outside the domain have a very different scaling
compared to those entries corresponding to interior nodes. We will
study the influence of this scaling effects on the convergence rate of
a multigrid algorithm for a 2d model problem.